To illustrate the definition of fitness network used in this work,
we need to build fitness landscapes instances.
For that, we use the well-known NK-model [1]}, [2]}, [3]}, recalling that for a genome of size \(N\) , the parameter \(K\in \lbrace  0,\ldots , N-1 \rbrace \)  corresponds to the epistatic coupling between loci and thus tunes the ruggedness of the landscape (for a brief introduction see Sec:Methods).
Here, we use two neighbourhood coupling models between loci: i) the adjacent model in which the \(K\)  loci are those closest to a focal locus \(x_i\) ; ii) the random neighbourhood model, where the \(K\)  loci are chosen randomly among the \(N-1\)  loci other than \(x_i\)  (an illustration of epistatic interactions is sketched in Sec:NK in Sec:Methods).
In Fig REF  we show representative examples of fitness networks engineered for \(N=4\)  with \(K\)  epistatic random neighbours (see REF  for the fitness networks with \(K\)  epistatic adjacent neighbours).
The landscapes range from single peaked \(K=0\)  (no epistatic interaction) to full rugged landscapes \(K=3\)  (highly connected epistatic interactions).
Global fitness maxima and minima are highlighted by encircled nodes.
In Fig REF A, REF B and REF C we show an instance of fitness networks for genotypes connected through pathways with point mutation steps.
They have the same topology as the genotype space \(\mathcal {Q}_{N=4}\)  and, therefore, are isomorphic to the graph representation of the fitness landscape \(\mathcal {F}\cong \mathrm {F}=(\mathcal {H}(4,2),f)\) .
When \(K=0\)  all the trajectories arrive to the single global maximum of fitness.
When we construct the fitness network with inversion mutations, we can verify in Fig REF D, REF E and REF F that there are more paths between all the genotypes, and so it is easier for an evolutionary process to explore more domains of the fitness landscape compared to point mutations.
Many of these paths connect genotypes such that \(d_H({\bf x},{\bf x}^{\prime })>1\) , and therefore, are like jumps between distant domains of the landscape.
We can also verify that, for a given fitness function \(f\) , a node that is a local optimum on the fitness graph \(\mathrm {F}=(\mathcal {H}(N,2),f)\) , is not necessarily
a local peak for inversion mutations in the fitness network.
In most of the cases, the fitness landscape is “smoothed out” by inversion mutations, since the notion of local peak fades in the fitness network.
However, the local peaks are not always smoothed out, as we can see in
Fig REF E for \(K=1\) , where genotype 0101 remains as a local peak. This is because 0101 cannot be mutated to genotype 1001 by any inversion mutation (see also the combinatorics of accessible mutants for 0101 in Fig REF ).
Note that for inversion mutations, it is verified that in some cases the global maximum can be reached from the global minimum in a single evolutionary step with \(d_H({\bf x},{\bf x}^{\prime })\ne 1\) , e.g. Fig REF E with \(d_H=2\) .
